Axiom ax-ac 196

Description: Defining property of the indefinite descriptor: it selects an element from any type. This is equivalent to global choice in ZF. (Contributed by Mario Carneiro, 10-Oct-2014.)

Assertion

Ref	Expression
ax-ac	|- T. |= (A.\p:(al -> *) (A.\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]))

Distinct variable group:   x,p

Detailed syntax breakdown of Axiom ax-ac

Step	Hyp	Ref	Expression
1		kt 8	. 2 term T.
2		tal 122	. . 3 term A.
3		hal	. . . . 5 type al
4		hb 3	. . . . 5 type *
5	3, 4	ht 2	. . . 4 type (al -> *)
6		vp	. . . 4 var p
7		vx	. . . . . 6 var x
8	5, 6	tv 1	. . . . . . . 8 term p:(al -> *)
9	3, 7	tv 1	. . . . . . . 8 term x:al
10	8, 9	kc 5	. . . . . . 7 term (p:(al -> *)x:al)
11		tat 191	. . . . . . . . 9 term @
12	11, 8	kc 5	. . . . . . . 8 term (@p:(al -> *))
13	8, 12	kc 5	. . . . . . 7 term (p:(al -> *)(@p:(al -> *)))
14		tim 121	. . . . . . 7 term ==>
15	10, 13, 14	kbr 9	. . . . . 6 term [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]
16	3, 7, 15	kl 6	. . . . 5 term \x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]
17	2, 16	kc 5	. . . 4 term (A.\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))])
18	5, 6, 17	kl 6	. . 3 term \p:(al -> *) (A.\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))])
19	2, 18	kc 5	. 2 term (A.\p:(al -> *) (A.\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]))
20	1, 19	wffMMJ2 11	1 wff T. |= (A.\p:(al -> *) (A.\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]))

This axiom is referenced by:  ac  197